Took me some time but I think I solved it
A. To find the area of region S, we need to integrate the difference between the functions f(x) and g(x) with respect to x, from 0 to 2. Since f(x) is defined piecewise, we need to split the integral into two parts:Area of region S = ∫(f(x) - g(x)) dx from x=0 to x=1 + ∫(f(x) - g(x)) dx from x=1 to x=2Evaluating each integral separately using the given functions:Area of region S = [(7/2) - (5/2)] dx from x=0 to x=1 + [(5/2) - (4/3)x^(3/2)] dx from x=1 to x=2
Area of region S = [1] dx from x=0 to x=1 + [(5/2) - (4/3)*2^(3/2)] - [(5/2) - (4/3)] dx from x=1 to x=2
Area of region S = 1 + [5/2 - (4/3)2^(3/2) + 4/3 - 5/2] = 2/3 + (5/2 - 4/32^(3/2))Therefore, the area of region S is approximately 1.302 units.
b. To find the volume of the solid generated when region S is revolved about the horizontal line, we need to use the disk method. Each cross-section perpendicular to the horizontal line is a disk with radius equal to the distance between the y-axis and the function f(x) minus the distance between the y-axis and the function g(x), and thickness dx.Volume of solid = π∫(f(x)^2 - g(x)^2) dx from x=0 to x=2Using the given functions f(x) and g(x):Volume of solid = π∫(7x - x^(3/2))^2 dx from x=0 to x=2
Volume of solid = π∫(49x^2 - 14x^(5/2) + x^3) dx from x=0 to x=2
Volume of solid = π[(49/3)2^3 - (14/7)(2^(7/2) - 1) + (1/4)*2^4]Therefore, the volume of the solid generated when region S is revolved about the horizontal line is approximately 59.542 units^3.
c. Each cross-section perpendicular to the x-axis is a rectangle with height equal to 7 times the length of its base in region S. Since the length of the base can be represented as (f(x) - g(x)), the area of each cross-section can be represented as 7(f(x) - g(x))dx. Therefore, the volume of the solid can be expressed as:Volume of solid = ∫7(f(x) - g(x)) dx from x=0 to x=2We already calculated the integral in part (a), so substituting that expression:Volume of solid = 7Area of region S
Volume of solid = 7(2/3 + 5/2 - 4/32^(3/2))
Volume of solid = 14/3 + 35/2 - 28/3*2^(3/2)Therefore, the integral expression for the volume of this solid is 14/3 + 35/2 - 28/3*2^(3/2) times the appropriate units^3.